There is just a bit of a terminology difference getting in the way here.
Yes, you can use the MC Handheld Sync functionality to sync your
media files to the SD card on your Android phone in exactly the same way you sync to your iPod now. The internal memory and the SD card installed appear as two separate devices in MC's Handheld Sync functionality.
JRiver for Android will then find the media files on your SD card and import them into its own Library. Then you can play the music.
As JRiver for Android uses its own Library I recommend that you make sure any tags you want used on the phone are saved to the files in MC on Windows. But most important ones are already saved to the file, such as Genre, Rating, and all the Artist, Album, etc. tags. Just make sure MC on Windows is set to write tags on file changes.
This is how I use MC with my Sony Xperia XZ Premium on Android Pie 9 now.
It works okay. Sometimes there are some little issues, such as I just noticed one of my Albums has all files duplicated. JRiver for Android doesn't have all the tools to allow me to work out why that happened, for example. It's importing all my photos on the phone as well, which isn't necessary. So it is still a work in progress.
But JRiver for Android works fine for what you want to do. It doesn't cost that much to buy and try out, either. Just do it!